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Scenario Generation based on Quantization

Scenario Generation based on Quantization. Moscow, September 2009. A. Pichler Department of Statistics and Decision Support Systems. Outline of this Talk. Approximation of Probability measures Quantization Trees, Scenario Generation. Comparison of Measures.

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Scenario Generation based on Quantization

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  1. Scenario Generationbased onQuantization Moscow, September 2009 A. Pichler Department of Statistics and Decision Support Systems

  2. Outline of this Talk • Approximation of Probability measures • Quantization • Trees, Scenario Generation

  3. Comparison of Measures Given 2 measures, how can we link them and assign a single number to account for distance?   maximum (comonotone) product measure minimum (antimonotone)

  4. Earth Mover Distance

  5. Distance for Measures As the initial space obeys a distance function, we thus may compute for every marginal distribution. ... is a metric, metrizing the weak topology.

  6. Summary cost function Distribution with adapted marginals is a metric on measures, metrizing the weak topology.

  7. Approximation • Proxy for Expectation reads • and convergence is to be understood in the weak (*) sense,

  8. Metrizable • The weak Topology is metrizable • Elaborated since 18th century G. Monge

  9. Outline of this Talk • Approximation of Probability Measures • Quantization • Trees, Scenario Generation

  10. Approximation • Approximation via Dirac Measures • Well defined?

  11. Discrete Situation original distribution Approximation

  12. Linking the Concepts Questions: • Where to place qQ? • How to choose the weights pq? ?

  13. Comparison of distances Example: Approximation of Gaussian Distribution 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 - 3 - 2 - 1 1 2 3 - 3 - 2 - 1 1 2 3 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 - 3 - 2 - 1 1 2 3 - 3 - 2 - 1 1 2 3

  14. Associated Region 1.0 0.8 1.2 1.0 0.6 0.8 0.4 0.6 0.4 0.2 0.2 - 3 - 2 - 1 1 2 3 - 3 - 2 - 1 1 2 3 Associated with given sample points are the intervals containing points being closest to the sample point.

  15. Voronoi Diagram Георгий Феодосьевич Вороной Борис Николаевич Делоне in 2 dimensions ... ... and in 3 dimensions 2 5 1 3 7 6 4 13 8 9 14 10 12 15 11 16 (c) Mathematica

  16. Lemma 1 Let with 2 then Let Xq be Voronoi and , then 3

  17. Solution to 1 ? Questions: • How to choose the weights pq? • Where to place qQ? Answer Part 1: • Compute the Voronoi Tessellation Xq and set • The even

  18. Towards Question 2 The problem of finding the best supporting nodes (Q) now transforms into the problem of minimizing the functional

  19. Lloyd Algorithm Lloyd Algorithm repeats 2 steps: • for Q, compute Xq and • for Xq, compute

  20. Example Optimal supporting nodes and Voronoi Diagram for the Bivariate Normal Distribution

  21. Selected Solutions Optimal supporting nodes are known in a few selected sitations only. They include (r=1) • Uniform distribution • Exponential distribution

  22. Towards Question 2 Both solutions have the form (for a function z) 1.2 1.0 0.8 0.6 0.4 0.2 - 3 - 2 - 1 1 2 3 Differentiating gives (R, r= 1) F... the cumulative distribution function, f... the density

  23. Asymptotic, motivation • For n increasing, the difference equation transforms into the differential equation

  24. again... If P has density ... ... then the supporting nodes have (asymptotic) density ... dimension As a consequence, tails get higher attention if r is chosen big enough.

  25. Solution to 2 ? Questions: • How to choose the weights pq? • Where to place qQ? Answer Part 2: • Solve related equations, or • use, for example, Lloyd Algorithm • Asymptotically, compute quantiles of

  26. Example Example: Approximation of Gaussian Distribution

  27. Theorem (Zador): The error of the best approximation satisfies Note: this theorem links P to a simple uniform distribution and discloses the rate of convergence.

  28. The Constant... ... has the following bounds cube proxy upper/ lower bound 0.7 0.6 0.5 0.4 0.3 0.2 (l2 Norm, r =1) dimension d 0.1 5 10 15 20 25 30

  29. Outline of this Talk • Approximation of Probability measures • Quantization • Trees, Scenario Generation

  30. A tree is a Quantizer Xt-1 Xt Xt+1 A tree is a special quantizer on the multidimensional space Assume, there exist stochastic kernels describing the transition

  31. Theorem Xt-1 Xt Xt+1 Suppose that then Note: the statement gives exact bounds if (independent). Note: May be used to explicitely construct a tree (scenario) with a distance given in advance.

  32. Outlook 2010+ > Scenario Reduction >> Römisch > Include Decisions at stage t >> Pflug > Applications and numerous implementations >> Hochreiter > Portfolio Optimization >> Kovacevic

  33. Thanks For Your Attention.

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